Why is the behaviour policy denoted by $\beta$ and the exploration policy by $ \mu'$ in the DDPG paper?
You are right, it is sloppy notation by the authors. However, the target network is not necessarily linked to the behaviour policy $\beta$ either.
Essentially when they take the expectation with respect to $\rho^\beta$ they are taking expectation with respect to a state distribution induced by some policy $\beta$ that is not necessarily the same as our current policy -- this is what makes DDPG an off-policy algorithm.
The target actor network $\mu'$ is used in the loss function for the critic; the target for the critic (ignoring parameters for brevity) $y = r + \gamma Q(s', \mu'(s'))$ where $s'$ is the state we transitioned to from $s$ when we took an action $a$.
Now, $a$ was sampled as part of the tuple $(s, a, r, s')$ from our replay buffer, meaning that $a$ will have been chosen according to some past version of our policy, plus some exploration noise $\mathcal{N}$. The way this links to the behaviour policy $\beta$ is that because we have sampled it from some old version of our policy, i.e. not our current policy, it is instead coming from this behaviour policy $\beta$.
The target network $\mu'$ is simply a copy of the current actor network where the weights are updated using the polyak averaging technique and is not really related to the behaviour policy $\beta$, at least not in any useful way for you to think about it.